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Euler's line

Source: 4-th Taiwanese Mathematical Olympiad 1995

January 16, 2007
geometrycircumcirclegeometry unsolved

Problem Statement

Let PP be a point on the circumcircle of a triangle A1A2A3A_{1}A_{2}A_{3}, and let HH be the orthocenter of the triangle. The feet B1,B2,B3B_{1},B_{2},B_{3} of the perpendiculars from PP to A2A3,A3A1,A1A2A_{2}A_{3},A_{3}A_{1},A_{1}A_{2} lie on a line. Prove that this line bisects the segment PHPH.
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